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pywafo/wafo/polynomial.py

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"""
Extended functions to operate on polynomials
"""
# ------------------------------------------------------------------------
# Name: polynomial
# Purpose: Functions to operate on polynomials.
#
# Author: pab
# polyXXX functions are based on functions found in the matlab toolbox polyutil
# written by
# Author: Peter J. Acklam
# E-mail: pjacklam@online.no
# WWW URL: http://home.online.no/~pjacklam
#
# Created: 30.12.2008
# Copyright: (c) pab 2008
# Licence: LGPL
# ------------------------------------------------------------------------
# !/usr/bin/env python
from __future__ import absolute_import
import warnings # @UnusedImport
from functools import reduce
from numpy.polynomial import polyutils as pu
from wafo.plotbackend import plotbackend as plt
import numpy as np
from numpy import (newaxis, arange, pi)
from scipy.fftpack import dct, idct as _idct
from numpy.lib.polynomial import * # @UnusedWildImport
from scipy.misc import pade # @UnresolvedImport
__all__ = np.lib.polynomial.__all__
__all__ = __all__ + ['pade', 'padefit', 'polyreloc', 'polyrescl', 'polytrim',
'poly2hstr', 'poly2str', 'polyshift', 'polyishift',
'map_from_intervall', 'map_to_intervall', 'cheb2poly',
'chebextr', 'chebroot', 'chebpoly', 'chebfit', 'chebval',
'chebder', 'chebint', 'Cheb1d', 'dct', 'idct']
def polyint(p, m=1, k=None):
"""
Return an antiderivative (indefinite integral) of a polynomial.
The returned order `m` antiderivative `P` of polynomial `p` satisfies
:math:`\\frac{d^m}{dx^m}P(x) = p(x)` and is defined up to `m - 1`
integration constants `k`. The constants determine the low-order
polynomial part
.. math:: \\frac{k_{m-1}}{0!} x^0 + \\ldots + \\frac{k_0}{(m-1)!}x^{m-1}
of `P` so that :math:`P^{(j)}(0) = k_{m-j-1}`.
Parameters
----------
p : {array_like, poly1d}
Polynomial to differentiate.
A sequence is interpreted as polynomial coefficients, see `poly1d`.
m : int, optional
Order of the antiderivative. (Default: 1)
k : {None, list of `m` scalars, scalar}, optional
Integration constants. They are given in the order of integration:
those corresponding to highest-order terms come first.
If ``None`` (default), all constants are assumed to be zero.
If `m = 1`, a single scalar can be given instead of a list.
See Also
--------
polyder : derivative of a polynomial
poly1d.integ : equivalent method
Examples
--------
The defining property of the antiderivative:
>>> import wafo.polynomial as wp
>>> p = wp.poly1d([1,1,1])
>>> P = wp.polyint(p)
>>> P
poly1d([ 0.33333333, 0.5 , 1. , 0. ])
>>> wp.polyder(P) == p
True
The integration constants default to zero, but can be specified:
>>> P = wp.polyint(p, 3)
>>> P(0)
0.0
>>> wp.polyder(P)(0)
0.0
>>> wp.polyder(P, 2)(0)
0.0
>>> P = wp.polyint(p, 3, k=[6, 5, 3])
>>> P
poly1d([ 0.01666667, 0.04166667, 0.16666667, 3. , 5. , 3. ])
Note that 3 = 6 / 2!, and that the constants are given in the order of
integrations. Constant of the highest-order polynomial term comes first:
>>> wp.polyder(P, 2)(0)
6.0
>>> wp.polyder(P, 1)(0)
5.0
>>> P(0)
3.0
"""
def _polyintnd(p, m, k):
ix = arange(len(p), 0, -1)
if p.ndim > 1:
ix = ix[..., newaxis]
pieces = p.shape[-1]
k0 = k[0] * np.ones((1, pieces), dtype=int)
else:
k0 = [k[0]]
y = np.concatenate((p.__truediv__(ix), k0), axis=0)
val = polyint(y, m - 1, k=k[1:])
if truepoly:
return poly1d(val)
return val
def _check_order(m):
if m < 0:
msg = "Order of integral must be positive (see polyder)"
raise ValueError(msg)
m = int(m)
_check_order(m)
if k is None:
k = np.zeros(m, float)
k = np.atleast_1d(k)
if len(k) == 1 and m > 1:
k = k[0] * np.ones(m, float)
if len(k) < m:
raise ValueError(
"k must be a scalar or a rank-1 array of length 1 or >m.")
truepoly = isinstance(p, poly1d)
p = np.asarray(p)
if m == 0:
if truepoly:
return poly1d(p)
return p
return _polyintnd(p, m, k)
def polyder(p, m=1):
"""
Return the derivative of the specified order of a polynomial.
Parameters
----------
p : poly1d or sequence
Polynomial to differentiate.
A sequence is interpreted as polynomial coefficients, see `poly1d`.
m : int, optional
Order of differentiation (default: 1)
Returns
-------
der : poly1d
A new polynomial representing the derivative.
See Also
--------
polyint : Anti-derivative of a polynomial.
poly1d : Class for one-dimensional polynomials.
Examples
--------
The derivative of the polynomial :math:`x^3 + x^2 + x^1 + 1` is:
>>> import wafo.polynomial as wp
>>> p = wp.poly1d([1,1,1,1])
>>> p2 = wp.polyder(p)
>>> p2
poly1d([3, 2, 1])
which evaluates to:
>>> p2(2.)
17.0
We can verify this, approximating the derivative with
``(f(x + h) - f(x))/h``:
>>> (p(2. + 0.001) - p(2.)) / 0.001
17.007000999997857
The fourth-order derivative of a 3rd-order polynomial is zero:
>>> wp.polyder(p, 2)
poly1d([6, 2])
>>> wp.polyder(p, 3)
poly1d([6])
>>> wp.polyder(p, 4)
poly1d([ 0.])
"""
def _polydernd(p, m):
n = len(p) - 1
ix = arange(n, 0, -1)
if p.ndim > 1:
ix = ix[..., newaxis]
y = ix * p[:-1]
val = polyder(y, m - 1)
if truepoly:
return poly1d(val)
return val
def _check_order(m):
if m < 0:
msg = "Order of derivative must be positive (see polyint)"
raise ValueError(msg)
m = int(m)
_check_order(m)
truepoly = isinstance(p, poly1d)
p = np.asarray(p)
if m == 0:
if truepoly:
return poly1d(p)
return p
return _polydernd(p, m)
def polydeg(x, y):
'''
Return optimal degree for polynomial fitting
N = POLYDEG(X,Y) finds the optimal degree for polynomial fitting,
according to the Akaike's information criterion.
Assuming that you want to find the degree N of a polynomial that fits
the data Y(X) best in a least-squares sense, the Akaike's information
criterion is defined by:
2*(N + 1) + n * (log(2 * pi * RSS / n) + 1)
where n is the number of points and RSS is the residual sum of squares.
The optimal degree N is defined here as that which minimizes AIC:
http://en.wikipedia.org/wiki/Akaike_Information_Criterion
Notes:
-----
If the number of data is small, POLYDEG may tend to return:
N = (number of points)-1.
ORTHOFIT is more appropriate than POLYFIT for polynomial fitting with
relatively high degrees.
Example:
-------
>>> import wafo.polynomial as wp
>>> x = np.linspace(0,10,300)
>>> noise = 0.05 * np.random.randn(x.size)
>>> noise = 0.05 * np.sin(100*x)
>>> y = np.sin(x ** 3 / 100) ** 2 + noise
>>> n = wp.polydeg(x,y)
>>> n
21
ys = wp.orthofit(x,y,n);
plt.plot(x, y, '.', x, ys, 'k')
See also
--------
polyfit, orthofit
'''
x, y = np.atleast_1d(x, y)
x = x.ravel()
y = y.ravel()
N = len(x)
# Search the optimal degree minimizing the Akaike's information criterion
# y(x) are fitted in a least-squares sense using a polynomial of degree n
# developed in a series of orthogonal polynomials.
ys = np.ones((N,)) * y.mean()
# correction for small sample sizes
logsum2 = (np.log(2 * pi * ((ys - y) ** 2).sum() / N) + 1)
AIC = 2 + N * logsum2 + 4 / (N - 2)
n = 1
nit = 0
# While-loop is stopped when a minimum is detected. 3 more steps are
# required to take AIC noise into account and to ensure that this minimum
# is a (likely) global minimum.
while nit < 8:
p = orthofit(x, y, n)
ys = orthoval(p, x)
# -- Akaike's Information Criterion
aic = (2 * (n + 1) * (1 + (n + 2) / (N - n - 2)) +
N * (np.log(2 * pi * np.sum((ys - y) ** 2) / N) + 1))
if aic >= AIC:
nit += 1
else:
nit = 0
AIC = aic
n = n + 1
if n >= N:
break
n = n - nit - 1
return n
def orthoval(p, x):
'''
Evaluation of orthogonal polynomial
Parameters
----------
p : array_like
2D array of polynomial coefficients (including coefficients equal
to zero) from highest degree to the constant term.
x : array_like
A number or a 1D array of numbers "at" which to evaluate `p`.
Returns
-------
values : ndarray
See Also
--------
orthofit
'''
p = np.atleast_2d(p)
n = p.shape[1] - 1
xi = np.atleast_1d(x)
shape0 = xi.shape
if n == 0:
return np.ones(shape0) * p[0]
xi = xi.ravel()
xn = np.ones((n + 1, len(xi)))
xn[1] = xi - p[1, 1]
for i in range(2, n + 1):
xn[i, :] = (xi - p[1, i]) * xn[i - 1, :] - p[2, i] * xn[i - 2, :]
ys = np.dot(p[0], xn)
return ys.reshape(shape0)
def ortho2poly(p):
"""
Converts orthogonal polynomial to ordinary polynomial coefficients
Parameters
----------
p : array-like
orthogonal polynomial coefficients
Returns
-------
p : ndarray
ordinary polynomial coefficients
It is not advised to do this for p.shape[1]>10 due to numerical
cancellations.
See also
--------
orthoval
orthofit
Examples
--------
>>> import numpy as np
>>> import wafo.polynomial as wp
>>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0])
>>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0])
>>> p = wp.orthofit(x, y, 3)
>>> p
array([[ 0. , -0.30285714, -0.16071429, 0.08703704],
[ 0. , 2.5 , 2.5 , 2.5 ],
[ 0. , 0. , 2.91666667, 2.13333333]])
>>> wp.ortho2poly(p)
array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254])
>>> wp.polyfit(x, y, 3)
array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254])
References
----------
"""
p = np.atleast_2d(p)
n = p.shape[1] - 1
if n == 0:
return p[0]
x = [1, ] * (n + 1)
x[1] = np.array([1, - p[1, 1]])
for i in range(2, n + 1):
x[i] = polyadd(polymul([1, - p[1, i]], x[i - 1]), - p[2, i] * x[i - 2])
for i in range(n + 1):
x[i] *= p[0, i]
return reduce(polyadd, x)
def orthofit(x, y, n):
'''
Fit orthogonal polynomial to data.
Parameters
---------
x, y : arrays
data Y(X) to fit to a polynomial.
n : integer
degree of fitted polynomial.
Returns
-------
p : array
orthogonal polynomial
Notes:
-----
Orthofit smooths/fits data using a polynomial of degree N developed in
a sequence of orthogonal polynomials. ORTHOFIT is more appropriate than
polyfit for polynomial fitting and smoothing since this method does not
involve any matrix linear system but a simple recursive procedure.
Degrees much higher than 30 could be used with orthogonal polynomials,
whereas badly conditioned matrices may appear with a classical
polynomial fitting of degree typically higher than 10.
To avoid using unnecessarily high degrees, you may let the function
POLYDEG choose it for you. POLYDEG finds an optimal polynomial degree
according to the Akaike's information criterion.
Example:
-------
>>> import wafo.polynomial as wp
>>> x = np.linspace(0,10,300);
>>> y = np.sin(x**3/100)**2 + 0.05*np.random.randn(x.size)
>>> p = wp.orthofit(x, y, 25)
>>> ys = wp.orthoval(p, x)
plot(x, y,'.',x, ys, 'k')
See also
--------
polydeg, polyfit, polyval
Reference:
---------
Methodes de calcul numerique 2. JP Nougier. Hermes Science
Publications, 2001. Section 4.7 pp 116-121
'''
x, y = np.atleast_1d(x, y)
x = x.ravel()
y = y.ravel()
# Particular case: n=0
if n == 0:
return y.mean()
# p = Coefficients of the orthogonal polynomials
p = np.zeros((3, n + 1))
p[1, 1] = x.mean()
N = len(x)
PL = np.ones((n + 1, N))
PL[1] = x - p[1, 1]
for i in range(2, n + 1):
p[1, i] = np.dot(x, PL[i - 1] ** 2) / np.sum(PL[i - 1] ** 2)
p[2, i] = np.dot(x, PL[i - 2] * PL[i - 1]) / np.sum(PL[i - 2] ** 2)
PL[i] = (x - p[1, i]) * PL[i - 1] - p[2, i] * PL[i - 2]
p[0, :] = np.dot(PL, y) / np.sum(PL ** 2, axis=1)
return p
# ys = np.dot(p[0, :], PL) # smoothed y
def polyreloc(p, x, y=0.0):
"""
Relocate polynomial
The polynomial `p` is relocated by "moving" it `x`
units along the x-axis and `y` units along the y-axis.
So the polynomial `r` is relative to the point (x,y) as
the polynomial `p` is relative to the point (0,0).
Parameters
----------
p : array-like, poly1d
vector or matrix of column vectors of polynomial coefficients to
relocate. (Polynomial coefficients are in decreasing order.)
x : scalar
distance to relocate P along x-axis
y : scalar
distance to relocate P along y-axis (default 0)
Returns
-------
r : ndarray, poly1d
vector/matrix/poly1d of relocated polynomial coefficients.
See also
--------
polyrescl
Example
-------
>>> import numpy as np
>>> import wafo.polynomial as wp
>>> p = np.arange(6); p.shape = (2,-1)
>>> wp.polyval(p,0)
array([3, 4, 5])
>>> wp.polyval(p,1)
array([3, 5, 7])
>>> r = polyreloc(p,-1) # move to the left along x-axis
>>> wp.polyval(r,-1) # = polyval(p,0)
array([3, 4, 5])
>>> wp.polyval(r,0) # = polyval(p,1)
array([3, 5, 7])
"""
truepoly = isinstance(p, poly1d)
r = np.atleast_1d(p).copy()
n = r.shape[0]
# Relocate polynomial using Horner's algorithm
for ii in range(n, 1, -1):
for i in range(1, ii):
r[i] = r[i] - x * r[i - 1]
r[-1] = r[-1] + y
if r.ndim > 1 and r.shape[-1] == 1:
r.shape = (r.size,)
if truepoly:
r = poly1d(r)
return r
def polyrescl(p, x, y=1.0):
"""
Rescale polynomial.
Parameters
----------
p : array-like, poly1d
vector or matrix of column vectors of polynomial coefficients to
rescale. (Polynomial coefficients are in decreasing order.)
x,y : scalars
defining the factors to rescale the polynomial `p` in
x-direction and y-direction, respectively.
Returns
-------
r : ndarray, poly1d
vector/matrix/poly1d of rescaled polynomial coefficients.
See also
--------
polyreloc
Example
-------
>>> import numpy as np
>>> import wafo.polynomial as wp
>>> p = np.arange(6); p.shape = (2,-1)
>>> wp.polyval(p,0)
array([3, 4, 5])
>>> wp.polyval(p,1)
array([3, 5, 7])
>>> r = wp.polyrescl(p,2) # scale by 2 along x-axis
>>> wp.polyval(r,0) # = polyval(p,0)
array([ 3., 4., 5.])
>>> wp.polyval(r,2) # = polyval(p,1)
array([ 3., 5., 7.])
"""
truepoly = isinstance(p, poly1d)
r = np.atleast_1d(p)
n = r.shape[0]
xscale = (float(x) ** arange(1 - n, 1))
if r.ndim == 1:
q = y * r * xscale
else:
q = y * r * xscale[:, newaxis]
if truepoly:
q = poly1d(q)
return q
def polytrim(p):
"""
Trim polynomial by stripping off leading zeros.
Parameters
----------
p : array-like, poly1d
vector or matrix of column vectors of polynomial coefficients in
decreasing order.
Returns
-------
r : ndarray, poly1d
vector/matrix/poly1d of trimmed polynomial coefficients.
Example
-------
>>> import wafo.polynomial as wp
>>> p = [0,1,2]
>>> wp.polytrim(p)
array([1, 2])
>>> p1 = [[0,0],[1,2],[3,4]]
>>> wp.polytrim(p1)
array([[1, 2],
[3, 4]])
"""
truepoly = isinstance(p, poly1d)
if truepoly:
return p
else:
r = np.atleast_1d(p).copy()
# Remove leading zeros
is_not_lead_zeros = np.logical_or.accumulate(r != 0, axis=0)
if r.ndim == 1:
r = r[is_not_lead_zeros]
else:
is_not_lead_zeros = np.any(is_not_lead_zeros, axis=1)
r = r[is_not_lead_zeros, :]
return r
def poly2hstr(p, variable='x'):
"""
Return polynomial as a Horner represented string.
Parameters
----------
p : array-like poly1d
vector of polynomial coefficients in decreasing order.
variable : string
display character for variable
Returns
-------
p_str : string
consisting of the polynomial coefficients in the vector P multiplied
by powers of the given `variable`.
Examples
--------
>>> import wafo.polynomial as wp
>>> wp.poly2hstr([1, 1, 2], 's' )
'(s + 1)*s + 2'
>>> wp.poly2hstr([-2, 1, 2, -1], 's' )
'((-2*s + 1)*s + 2)*s - 1'
>>> wp.poly2hstr([2, 0, 2, 1], 's' )
'(2*s**2 + 2)*s + 1'
>>> wp.poly2hstr([0], 's' )
'0'
See also
--------
poly2str
"""
def _append_coef(s, coef, expon, var):
# Is it the first term?
isfirst = s == ''
# Add sign, but we don't need a leading plus-sign.
if isfirst:
if coef < 0:
s = '-' # Unary minus.
else:
sgn = '-' if coef < 0 else '+'
s = '{0:s} {1:s} '.format(s, sgn)
# We need the coefficient only if it is different from 1 or -1 or
# when it is the constant term.
needcoef = ((abs(coef) != 1) | (expon == 0) & isfirst) | 1 - isfirst
# Append the coefficient if it is different from one or when it is
# the constant term.
if needcoef:
s = '{0:s}{1:.20g}'.format(s, abs(coef))
# We need the variable except in the constant term.
needvar = expon != 0
# Append variable if necessary.
if needvar:
# Append a multiplication sign if necessary.
if needcoef:
if 1 - isfirst:
s = '({0:s})'.format(s)
s = '{0:s}*'.format(s)
s = '{0:s}{1:s}'.format(s, var)
return s
var = variable
coefs = polytrim(np.atleast_1d(p))
order = len(coefs) - 1 # Order of polynomial.
s = '' # Initialize output string.
ix = 1
for expon in range(order, -1, -1):
coef = coefs[order - expon]
# There is no point in adding a zero term (except if it's the only
# term, but we'll take care of that later).
if coef == 0:
ix += 1
else:
# Append exponent if necessary.
if ix > 1:
s = '{0:s}**{1:d}'.format(s, ix)
ix = 1
s = _append_coef(s, coef, expon, var)
# Now treat the special case where the polynomial is zero.
if s == '':
s = '0'
return s
def poly2str(p, variable='x'):
"""
Return polynomial as a string.
Parameters
----------
p : array-like poly1d
vector of polynomial coefficients in decreasing order.
variable : string
display character for variable
Returns
-------
p_str : string
consisting of the polynomial coefficients in the vector P multiplied
by powers of the given `variable`.
See also
--------
poly2hstr
Examples
--------
>>> import wafo.polynomial as wp
>>> wp.poly2str([1, 1, 2], 's' )
's**2 + s + 2'
>>> wp.poly2str([-2, 1, 2, 0, 0], 's' )
'-2*s**4 + s**3 + 2*s**2'
>>> wp.poly2hstr([0], 's' )
'0'
"""
def _coefstr_0(coefstr, k):
if coefstr != '0':
newstr = '{0:s}'.format(coefstr)
else:
newstr = '0' if k == 0 else ''
return newstr
def _coefstr_1(coefstr, var):
if coefstr == '0':
newstr = ''
elif coefstr in ['b', '1']:
newstr = var
else:
newstr = '{0:s}*{1:s}'.format(coefstr, var)
return newstr
def _coefstr_n(coefstr, var, power):
if coefstr == '0':
newstr = ''
elif coefstr in ['b', '1']:
newstr = '{0:s}**{1:d}'.format(var, power)
else:
newstr = '{0:s}*{1:s}**{2:d}'.format(coefstr, var, power)
return newstr
def _add_strings(thestr, newstr, ck, k):
if k > 0:
if newstr != '':
sgn = '-' if ck < 0 else '+'
thestr = "{0:s} {1:s} {2:s}".format(thestr, sgn, newstr)
elif (k == 0) and (newstr != '') and (ck < 0):
thestr = "-{0:s}".format(newstr)
else:
thestr = newstr
return thestr
thestr = "0"
var = variable
# Remove leading zeros
coeffs = polytrim(np.atleast_1d(p))
N = len(coeffs) - 1
for k, ck in enumerate(coeffs):
coefstr = '%.4g' % abs(ck)
if coefstr[-4:] == '0000':
coefstr = coefstr[:-5]
power = (N - k)
if power == 0:
newstr = _coefstr_0(coefstr, k)
elif power == 1:
newstr = _coefstr_1(coefstr, var)
else:
newstr = _coefstr_n(coefstr, var, power)
thestr = _add_strings(thestr, newstr, ck, k)
return thestr
def polyshift(py, a=-1, b=1):
"""
Polynomial coefficient shift
Polyshift shift the polynomial coefficients by a variable shift:
Y = 2*(X-.5*(b+a))/(b-a)
i.e., the interval -1 <= Y <= 1 is mapped to the interval a <= X <= b
Parameters
----------
py : array-like
polynomial coefficients for the variable y.
a,b : scalars
lower and upper limit.
Returns
-------
px : ndarray
polynomial coefficients for the variable x.
See also
--------
polyishift
Example
-------
>>> import wafo.polynomial as wp
>>> py = [1, 0]
>>> px = wp.polyshift(py,0,5)
>>> wp.polyval(px,[0, 2.5, 5]) #% This is the same as the line below
array([-1., 0., 1.])
>>> wp.polyval(py,[-1, 0, 1 ])
array([-1, 0, 1])
"""
if (a == -1) & (b == 1):
return py
L = b - a
return polyishift(py, -(2. + b + a) / L, (2. - b - a) / L)
def polyishift(px, a=-1, b=1):
"""
Inverse polynomial coefficient shift
Polyishift does the inverse of Polyshift,
shift the polynomial coefficients by a variable shift:
Y = 2*(X-.5*(b+a)/(b-a)
i.e., the interval a <= X <= b is mapped to the interval -1 <= Y <= 1
Parameters
----------
px : array-like
polynomial coefficients for the variable x.
a,b : scalars
lower and upper limit.
Returns
-------
py : ndarray
polynomial coefficients for the variable y.
See also
--------
polyishift
Example
-------
>>> import wafo.polynomial as wp
>>> px = [1, 0]
>>> py = wp.polyishift(px,0,5);
>>> wp.polyval(px,[0, 2.5, 5]) #% This is the same as the line below
array([ 0. , 2.5, 5. ])
>>> wp.polyval(py,[-1, 0, 1])
array([ 0. , 2.5, 5. ])
"""
if (a == -1) & (b == 1):
return px
L = b - a
xscale = 2. / L
xloc = -float(a + b) / L
return polyreloc(polyrescl(px, xscale), xloc)
def map_from_interval(x, a, b):
"""F(x), where F: [a,b] -> [-1,1]."""
return (x - (b + a) / 2.0) * (2.0 / (b - a))
def map_to_interval(x, a, b):
"""F(x), where F: [-1,1] -> [a,b]."""
return (x * (b - a) + (b + a)) / 2.0
def poly2cheb(p, a=-1, b=1):
"""
Convert polynomial coefficients into Chebyshev coefficients
Parameters
----------
p : array-like
polynomial coefficients
a,b : real scalars
lower and upper limits (Default -1,1)
Returns
-------
ck : ndarray
Chebychef coefficients
POLY2CHEB do the inverse of CHEB2POLY: given a vector of polynomial
coefficients AK, returns an equivalent vector of Chebyshev
coefficients CK.
This is useful for economization of power series.
The steps for doing so:
1. Convert polynomial coefficients to Chebychev coefficients, CK.
2. Truncate the CK series to a smaller number of terms, using the
coefficient of the first neglected Chebychev polynomial as an error
estimate.
3 Convert back to a polynomial by CHEB2POLY
See also
--------
cheb2poly
chebval
chebfit
Examples
--------
>>> import numpy as np
>>> import wafo.polynomial as wp
>>> p = np.arange(5)
>>> ck = wp.poly2cheb(p)
>>> wp.cheb2poly(ck)
array([ 1., 2., 3., 4.])
Reference
---------
William H. Press, Saul Teukolsky,
William T. Wetterling and Brian P. Flannery (1997)
"Numerical recipes in Fortran 77", Vol. 1, pp 184-194
"""
f = poly1d(p)
n = len(f.coeffs)
return chebfit(f, n, a, b)
def cheb2poly(ck, a=-1, b=1):
"""
Converts Chebyshev coefficients to polynomial coefficients
Parameters
----------
ck : array-like
Chebychef coefficients
a,b : real, scalars
lower and upper limits (Default -1,1)
Returns
-------
p : ndarray
polynomial coefficients
It is not advised to do this for len(ck)>10 due to numerical cancellations.
See also
--------
chebval
chebfit
Examples
--------
>>> import numpy as np
>>> import wafo.polynomial as wp
>>> p = np.arange(5)
>>> ck = wp.poly2cheb(p)
>>> wp.cheb2poly(ck)
array([ 1., 2., 3., 4.])
References
----------
http://en.wikipedia.org/wiki/Chebyshev_polynomials
http://en.wikipedia.org/wiki/Chebyshev_form
http://en.wikipedia.org/wiki/Clenshaw_algorithm
"""
n = len(ck)
b_Nmi = np.zeros(1)
b_Nmip1 = np.zeros(1)
y = np.r_[2 / (b - a), -(a + b) / (b - a)]
y2 = 2. * y
# Clenshaw recurence
for ix in range(n - 1):
tmp = b_Nmi
b_Nmi = polymul(y2, b_Nmi) # polynomial multiplication
nb = len(b_Nmip1)
b_Nmip1[-1] = b_Nmip1[-1] - ck[ix]
b_Nmi[-nb::] = b_Nmi[-nb::] - b_Nmip1
b_Nmip1 = tmp
p = polymul(y, b_Nmi) # polynomial multiplication
nb = len(b_Nmip1)
b_Nmip1[-1] = b_Nmip1[-1] - ck[n - 1]
p[-nb::] = p[-nb::] - b_Nmip1
return polytrim(p)
def chebextr(n):
"""
Return roots of derivative of Chebychev polynomial of the first kind.
All local extreme values of the polynomial are either -1 or 1. So,
CHEBPOLY( N, CHEBEXTR(N) ) ) return the same as (-1).^(N:-1:0)
except for the numerical noise in the former.
Because the extreme values of Chebychev polynomials of the first
kind are either -1 or 1, their roots are often used as starting
values for the nodes in minimax approximations.
Parameters
----------
n : scalar, integer
degree of Chebychev polynomial.
Examples
--------
>>> import wafo.polynomial as wp
>>> x = wp.chebextr(4)
>>> wp.chebpoly(4,x)
array([ 1., -1., 1., -1., 1.])
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_nodes
http://en.wikipedia.org/wiki/Chebyshev_polynomials
"""
return - np.cos((pi * arange(n + 1)) / n)
def chebroot(n, kind=1):
"""
Return roots of Chebychev polynomial of the first or second kind.
The roots of the Chebychev polynomial of the first kind form a particularly
good set of nodes for polynomial interpolation because the resulting
interpolation polynomial minimizes the problem of Runge's phenomenon.
Parameters
----------
n : scalar, integer
degree of Chebychev polynomial.
kind: 1 or 2, optional
kind of Chebychev polynomial (default 1)
Examples
--------
>>> import numpy as np
>>> import wafo.polynomial as wp
>>> x = wp.chebroot(3)
>>> np.allclose(wp.chebpoly(3,x), [0, 0, 0])
True
>>> wp.chebpoly(3)
array([ 4., 0., -3., 0.])
>>> x2 = wp.chebroot(4, kind=2)
>>> np.allclose(wp.chebpoly(4,x2,kind=2), [0, 0, 0, 0])
True
>>> wp.chebpoly(4,kind=2)
array([ 16., 0., -12., 0., 1.])
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_nodes
http://en.wikipedia.org/wiki/Chebyshev_polynomials
"""
if kind not in (1, 2):
raise ValueError('kind must be 1 or 2')
return - np.cos(pi * (arange(n) + 0.5 * kind) / (n + kind - 1))
def chebpoly(n, x=None, kind=1):
"""
Return Chebyshev polynomial of the first or second kind.
These polynomials are orthogonal on the interval [-1,1], with
respect to the weight function w(x) = (1-x**2)**(-1/2+kind-1).
chebpoly(n) returns coefficients of the Chebychev polynomial of degree N.
chebpoly(n,x) returns the Chebychev polynomial of degree N evaluated at X.
Parameters
----------
n : integer, scalar
degree of Chebychev polynomial.
x : array-like, optional
evaluation points
kind: 1 or 2, optional
kind of Chebychev polynomial (default 1)
Returns
-------
p : ndarray
polynomial coefficients if x is None.
Chebyshev polynomial evaluated at x otherwise
Examples
--------
>>> import numpy as np
>>> import wafo.polynomial as wp
>>> x = wp.chebroot(3)
>>> np.allclose(wp.chebpoly(3,x), [0, 0, 0])
True
>>> wp.chebpoly(3)
array([ 4., 0., -3., 0.])
>>> x2 = wp.chebroot(4,kind=2)
>>> np.allclose(wp.chebpoly(4,x2,kind=2), [0, 0, 0, 0])
True
>>> wp.chebpoly(4,kind=2)
array([ 16., 0., -12., 0., 1.])
>>> wp.chebpoly(0,kind=2)
array([ 1.])
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_polynomials
"""
if x is None: # Calculate coefficients.
if n == 0:
p = np.ones(1)
else:
p = np.round(pow(2, n - 2 + kind) * poly(chebroot(n, kind=kind)))
p[1::2] = 0
return p
else: # Evaluate polynomial in chebychev form
ck = np.zeros(n + 1)
ck[0] = 1.
return _chebval(np.atleast_1d(x), ck, kind=kind)
def chebfit(fun, n=10, a=-1, b=1, trace=False):
"""
Computes the Chebyshevs coefficients
so that f(x) can be approximated by:
n-1
f(x) = sum ck*Tk(x)
k=0
where Tk is the k'th Chebyshev polynomial of the first kind.
Parameters
----------
fun : callable
function to approximate
n : integer, scalar, optional
number of base points (abscissas). Default n=10 (maximum 50)
a,b : real, scalars, optional
integration limits
Returns
-------
ck : ndarray
polynomial coefficients in Chebychev form.
Examples
--------
Fit exp(x)
>>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> a = 0; b = 2
>>> ck = wp.chebfit(np.exp,7,a,b);
>>> x = np.linspace(0,4);
>>> x1 = wp.chebroot(9)*(b-a)/2+(b+a)/2
>>> ck1 = wp.chebfit(np.exp(x1))
h=plt.plot(x, np.exp(x), 'r', x, wp.chebval(x,ck,a,b), 'g.')
h = plt.plot(x,np.exp(x), 'r', x, wp.chebval(x,ck1,a,b),'g.')
plt.close()
See also
--------
chebval
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_nodes
http://mathworld.wolfram.com/ChebyshevApproximationFormula.html
W. Fraser (1965)
"A Survey of Methods of Computing Minimax and Near-Minimax Polynomial
Approximations for Functions of a Single Independent Variable"
Journal of the ACM (JACM), Vol. 12 , Issue 3, pp 295 - 314
"""
if (n > 50):
warnings.warn('CHEBFIT should only be used for n<50')
if hasattr(fun, '__call__'):
x = map_to_interval(chebroot(n), a, b)
f = fun(x)
if trace:
plt.plot(x, f, '+')
else:
f = fun
n = len(f)
# N-1
# c[k] = (2/N) sum w[n] f[n]*cos(pi*k*(2n+1)/(2N)), 0 <= k < N.
# n=0
#
# w[0] = 0.5, w[n]=1 for n>0
ck = dct(f[::-1]) / n
ck[0] = ck[0] / 2.
return ck[::-1]
def chebfit_dct(f, n=(10, ), domain=None):
"""
Fit Chebyshev series to N-dimensional function
so that f(x1, x2,..., xn) can be approximated by:
.. math:: f(x_1, x_2,...,x_n) =
\\sum_{i,j,...k} c_i T_i(x_1)*...*c_k T_k(x_n) ,
where Tk is the k'th Chebyshev polynomial of the first kind.
Parameters
----------
f : callable
function to approximate
n : list of integers, optional
number of base points (abscissas) used for each dimension.
Default n=10 (maximum 50)
domain : vector [a1,b1,a2,b2 ,..., an, bn], optional
defining the rectangle [a1, b1] x [a2, b2] x ...x [an, bn].
(default domain = (-1,1) * len(n))
Returns
-------
ck : ndarray
polynomial coefficients in Chebychev form.
Examples
--------
Fit exponential function
>>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> domain = (0, 2)
>>> ck = wp.chebfit_dct(np.exp, 7, domain)
>>> np.allclose(ck, [3.44152387e+00, 3.07252345e+00, 7.38000848e-01,
... 1.20520053e-01, 1.48805268e-02, 1.47579673e-03,
... 1.21719524e-04])
True
>>> x1 = wp.map_to_interval(wp.chebroot(9), *domain)
>>> ck1 = wp.chebfit(np.exp(x1))
>>> np.allclose(ck1, [5.40019009e-07, 8.69418381e-06, 1.22261037e-04,
... 1.47582673e-03, 1.48805283e-02, 1.20520053e-01,
... 7.38000848e-01, 3.07252345e+00, 3.44152387e+00])
True
x = np.linspace(0,4)
h = plt.plot(x, np.exp(x), 'r', x, wp.chebvalnd(ck, x,ck,a,b), 'g.')
h = plt.plot(x, np.exp(x), 'r', x, wp.chebvalnd(ck1, x,ck1,a,b),'b.')
plt.close()
See also
--------
chebval, chebvalnd
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_nodes
http://mathworld.wolfram.com/ChebyshevApproximationFormula.html
W. Fraser (1965)
"A Survey of Methods of Computing Minimax and Near-Minimax Polynomial
Approximations for Functions of a Single Independent Variable"
Journal of the ACM (JACM), Vol. 12 , Issue 3, pp 295 - 314
"""
n = np.atleast_1d(n)
if np.any(n > 50):
warnings.warn('CHEBFIT should only be used for n<50')
if hasattr(f, '__call__'):
if domain is None:
domain = (-1, 1) * len(n)
domain = np.atleast_2d(domain).reshape((-1, 2))
xi = [map_to_interval(chebroot(ni), d[0], d[1])
for ni, d in zip(n, domain)]
Xi = np.meshgrid(*xi)
ck = f(*Xi)
else:
ck = f
n = ck.shape
ndim = len(n)
for i in range(ndim):
ck = dct(ck[..., ::-1])
ck[..., 0] = ck[..., 0] / 2.
if i < ndim-1:
ck = np.rollaxis(ck, axis=-1)
return ck / np.product(n)
def idct(x, n=None):
"""
Inverse Discrete Cosine Transform
N-1
x[k] = 1/N sum w[n]*y[n]*cos(pi*k*(2n+1)/(2*N)), 0 <= k < N.
n=0
w(0) = 1/2
w(n) = 1 for n>0
Examples
--------
>>> import numpy as np
>>> import wafo.polynomial as wp
>>> x = np.arange(5)*1.0
>>> np.allclose(idct(dct(x)), x)
True
>>> np.allclose(dct(idct(x)), x)
True
Reference
---------
http://en.wikipedia.org/wiki/Discrete_cosine_transform
http://users.ece.utexas.edu/~bevans/courses/ee381k/lectures/
"""
return _idct(x, n=n, norm=None)*0.5/len(x)
def _chebval(x, ck, kind=1):
"""
Evaluate polynomial in Chebyshev form.
A polynomial of degree N in Chebyshev form is a polynomial p(x):
N
p(x) = sum ck*Tk(x)
k=0
or
N
p(x) = sum ck*Uk(x)
k=0
where Tk and Uk are the k'th Chebyshev polynomial of the first and second
kind, respectively.
References
----------
http://en.wikipedia.org/wiki/Clenshaw_algorithm
http://mathworld.wolfram.com/ClenshawRecurrenceFormula.html
"""
n = len(ck)
b_Nmi = np.zeros(x.shape) # b_(N-i)
b_Nmip1 = b_Nmi.copy() # b_(N-i+1)
x2 = 2 * x
# Clenshaw reccurence
for ix in range(n - 1):
tmp = b_Nmi
b_Nmi = x2 * b_Nmi - b_Nmip1 + ck[ix]
b_Nmip1 = tmp
return kind * x * b_Nmi - b_Nmip1 + ck[n - 1]
def chebval(x, ck, a=-1, b=1, kind=1, fill=None):
"""
Evaluate polynomial in Chebyshev form at X
A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form
N
p(x) = sum ck*Tk(x)
k=0
where Tk is the k'th Chebyshev polynomial of the first or second kind.
Paramaters
----------
x : array-like
points to evaluate
ck : array-like
polynomial coefficients in Chebyshev form ordered from highest degree
to zero
a,b : real, scalars, optional
limits for polynomial (Default -1,1)
kind: 1 or 2, optional
kind of Chebychev polynomial (default 1)
fill : scalar, optional
If provided, define value to return for `x < a` or `b < x`.
Examples
--------
Plot Chebychev polynomial of the first kind and order 4:
>>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> x = np.linspace(-1,1)
>>> ck = np.zeros(5); ck[-1]=1
>>> y = wp.chebval(x,ck)
h = plt.plot(x, y, x, wp.chebpoly(4,x),'.')
plt.close()
Fit exponential function:
>>> import matplotlib.pyplot as plt
>>> ck = wp.chebfit(np.exp,7,0,2)
>>> x = np.linspace(0,4);
>>> y2 = wp.chebval(x,ck,0,2)
h=plt.plot(x, y2, 'g', x, np.exp(x))
plt.close()
See also
--------
chebfit
References
----------
http://en.wikipedia.org/wiki/Clenshaw_algorithm
http://mathworld.wolfram.com/ClenshawRecurrenceFormula.html
"""
y = map_from_interval(np.atleast_1d(x), a, b)
if fill is None:
f = _chebval(y, ck, kind=kind)
else:
cond = (abs(y) <= 1)
f = np.where(cond, 0, fill)
if np.any(cond):
yk = np.extract(cond, y)
f[cond] = _chebval(yk, ck, kind=kind)
return f
def chebder(ck, a=-1, b=1):
"""
Differentiate Chebyshev polynomial
Parameters
----------
ck : array-like
polynomial coefficients in Chebyshev form of function to differentiate
a,b : real, scalars
limits for polynomial(Default -1,1)
Return
------
cder : ndarray
polynomial coefficients in Chebyshev form of the derivative
Examples
--------
Fit exponential function:
>>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> ck = wp.chebfit(np.exp,7,0,2)
>>> x = np.linspace(0,4)
>>> ck2 = wp.chebder(ck,0,2)
>>> y = wp.chebval(x,ck2,0,2)
h = plt.plot(x, y, 'g', x, np.exp(x), 'r')
plt.close()
See also
--------
chebint
chebfit
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_polynomials
W. Fraser (1965)
"A Survey of Methods of Computing Minimax and Near-Minimax Polynomial
Approximations for Functions of a Single Independent Variable"
Journal of the ACM (JACM), Vol. 12 , Issue 3, pp 295 - 314
"""
n = len(ck) - 1
cder = np.zeros(n, dtype=np.asarray(ck).dtype)
cder[0] = 2 * n * ck[0]
cder[1] = 2 * (n - 1) * ck[1]
for j in range(2, n):
cder[j] = cder[j - 2] + 2 * (n - j) * ck[j]
return cder * 2. / (b - a) # Normalize to the interval b-a.
def chebint(ck, a=-1, b=1):
"""
Integrate Chebyshev polynomial
Parameters
----------
ck : array-like
polynomial coefficients in Chebyshev form of function to integrate.
a,b : real, scalars
limits for polynomial(Default -1,1)
Return
------
cint : ndarray
polynomial coefficients in Chebyshev form of the integrated function
Examples
--------
Fit exponential function:
>>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> ck = wp.chebfit(np.exp, 7, 0, 2)
>>> x = np.linspace(0,4)
>>> ck2 = wp.chebint(ck, 0, 2);
>>> y = wp.chebval(x, ck2, 0, 2)
h = plt.plot(x, y, 'g', x, np.exp(x), 'r.')
plt.close()
See also
--------
chebder
chebfit
Reference
---------
http://en.wikipedia.org/wiki/Chebyshev_polynomials
W. Fraser (1965)
"A Survey of Methods of Computing Minimax and Near-Minimax Polynomial
Approximations for Functions of a Single Independent Variable"
Journal of the ACM (JACM), Vol. 12 , Issue 3, pp 295 - 314
"""
# int T0(x) = T1(x)+1
# int T1(x) = 0.5*(T2(x)/2-T0/2)
# int Tn(x) dx = 0.5*{Tn+1(x)/(n+1) - Tn-1(x)/(n-1)}
# N
# p(x) = sum cn*Tn(x)
# n=0
# int p(x) dx = sum cn * int(Tn(x)dx) =
# 0.5*sum cn *{Tn+1(x)/(n+1) - Tn-1(x)/(n-1)} = 0.5 sum (cn-1-cn+1)*Tn/n n>0
n = len(ck)
cint = np.zeros(n)
con = 0.25 * (b - a)
dif1 = np.diff(ck[-1::-2])
ix1 = np.r_[1:n - 1:2]
cint[ix1] = -(con * dif1) / ix1
if n > 3:
dif2 = np.diff(ck[-2::-2])
ix2 = np.r_[2:n - 1:2]
cint[ix2] = -(con * dif2) / ix2
cint = cint[::-1]
# cint(n) is a special case
cint[-1] = (con * ck[n - 2]) / (n - 1)
# Set integration constant
cint[0] = 2 * np.sum((-1) ** np.r_[0:n - 1] * cint[-2::-1])
return cint
class Cheb1d(object):
coeffs = None
order = None
a = None
b = None
kind = None
def __init__(self, ck, a=-1, b=1, kind=1):
if isinstance(ck, Cheb1d):
for key in ck.__dict__.keys():
self.__dict__[key] = ck.__dict__[key]
return
cki = trim_zeros(np.atleast_1d(ck), 'b')
if len(cki.shape) > 1:
raise ValueError("Polynomial must be 1d only.")
self.__dict__['coeffs'] = cki
self.__dict__['order'] = len(cki) - 1
self.__dict__['a'] = a
self.__dict__['b'] = b
self.__dict__['kind'] = kind
def __call__(self, x):
return chebval(x, self.coeffs, self.a, self.b, self.kind)
def __array__(self, t=None):
if t:
return np.asarray(self.coeffs, t)
else:
return np.asarray(self.coeffs)
def __repr__(self):
vals = repr(self.coeffs)
vals = vals[6:-1]
return "Cheb1d(%s)" % vals
def __len__(self):
return self.order
def __str__(self):
pass
def __neg__(self):
new = Cheb1d(self)
new.coeffs = -self.coeffs
return new
def __pos__(self):
return self
def __add__(self, other):
other = Cheb1d(other)
new = Cheb1d(self)
new.coeffs = polyadd(self.coeffs, other.coeffs)
return new
def __radd__(self, other):
return self.__add__(other)
def __sub__(self, other):
other = Cheb1d(other)
new = Cheb1d(self)
new.coeffs = polysub(self.coeffs, other.coeffs)
return new
def __rsub__(self, other):
other = Cheb1d(other)
new = Cheb1d(self)
new.coeffs = polysub(other.coeffs, new.coeffs)
return new
def __eq__(self, other):
other = Cheb1d(other)
return (np.all(self.coeffs == other.coeffs) and (self.a == other.a) and
(self.b == other.b) and (self.kind == other.kind))
def __ne__(self, other):
return np.any(self.coeffs != other.coeffs) or (self.a != other.a) or (
self.b != other.b) or (self.kind != other.kind)
def __setattr__(self, key, val):
raise ValueError("Attributes cannot be changed this way.")
def __getattr__(self, key):
if key in ['c', 'coef', 'coefficients']:
return self.coeffs
elif key in ['o']:
return self.order
elif key in ['a']:
return self.a
elif key in ['b']:
return self.b
elif key in ['k']:
return self.kind
else:
try:
return self.__dict__[key]
except KeyError:
raise AttributeError(
"'%s' has no attribute '%s'" %
(self.__class__, key))
def __getitem__(self, val):
if val > self.order:
return 0
if val < 0:
return 0
return self.coeffs[val]
def __setitem__(self, key, val):
# ind = self.order - key
if key < 0:
raise ValueError("Does not support negative powers.")
if key > self.order:
zr = np.zeros(key - self.order, self.coeffs.dtype)
self.__dict__['coeffs'] = np.concatenate((self.coeffs, zr))
self.__dict__['order'] = key
self.__dict__['coeffs'][key] = val
return
def __iter__(self):
return iter(self.coeffs)
def integ(self, m=1):
"""
Return an antiderivative (indefinite integral) of this polynomial.
Refer to `chebint` for full documentation.
See Also
--------
chebint : equivalent function
"""
integ = Cheb1d(self)
integ.coeffs = chebint(self.coeffs, self.a, self.b)
return integ
def deriv(self, m=1):
"""
Return a derivative of this polynomial.
Refer to `chebder` for full documentation.
See Also
--------
chebder : equivalent function
"""
der = Cheb1d(self)
der.coeffs = chebder(self.coeffs, self.a, self.b)
return der
def padefit(c, m=None):
"""
Rational polynomial fitting from polynomial coefficients
Parameters
----------
c : array-like
coefficients of power series expansion from highest degree to zero.
m : scalar integer
order of denominator polynomial. (Default floor((len(c)-1)/2))
Returns
-------
num, den : poly1d
numerator and denominator polynomials for the pade approximation
If the function is well approximated by
M+N+1
f(x) = sum c(2*n+2-k)*x^k
k=0
then the pade approximation is given by
M
sum c1(n-k+1)*x^k
k=0
f(x) = ------------------------
N
sum c2(n-k+1)*x^k
k=0
Note: c must be ordered for direct use with polyval
Example
-------
Pade approximation to exp(x)
>>> import scipy.special as sp
>>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> c = wp.poly1d(1./sp.gamma(np.r_[6+1:0:-1]))
>>> [p, q] = wp.padefit(c)
>>> p; q
poly1d([ 0.00277778, 0.03333333, 0.2 , 0.66666667, 1. ])
poly1d([ 0.03333333, -0.33333333, 1. ])
x = np.linspace(0,4)
h = plt.plot(x, c(x), x, p(x)/q(x), 'g-', x,np.exp(x), 'r.')
plt.close()
See also
--------
scipy.misc.pade
"""
if not m:
m = int(np.floor((len(c) - 1) * 0.5))
c = np.asarray(c)
return pade(c[::-1], m)
def test_pade():
cof = np.array(([1.0, 1.0, 1.0 / 2, 1. / 6, 1. / 24]))
p, q = pade(cof, 2)
t = arange(0, 2, 0.1)
assert(np.all(abs(p(t) / q(t) - np.exp(t)) < 0.3))
def padefitlsq(fun, m, k, a=-1, b=1, trace=False, x=None, end_points=True):
"""
Rational polynomial fitting. A minimax solution by least squares.
Parameters
----------
fun : callable or a vector
function to approximate. If fun and x are supplied as vectors the
vectors must satisfy: len(fun)=len(x) > (m+k+1)*8.
m, k : integer
number of coefficients of the numerator and denominater, respectively.
a, b : real scalars
evaluation limits, (default a=-1,b=1)
trace : bool
if True plot values and fitted function.
end_points : bool
if True x = chebextr(npt - 1)
otherwise x = chebroot(npt, kind=1).
Note set end_points to True if there are singularities close to the
endpoints.
Returns
-------
num, den : poly1d
numerator and denominator polynomials for the pade approximation
dev : ndarray
maximum absolute deviation of the approximation
The pade approximation is given by
m
sum c1[m-i]*x**i
i=0
f(x) = ------------------------
k
sum c2[k-i]*x**i
i=0
Note: c1 and c2 are ordered for direct use with polyval
Example
-------
Pade approximation to exp(x) between 0 and 2
>>> import matplotlib.pyplot as plt
>>> import wafo.polynomial as wp
>>> [c1, c2] = wp.padefitlsq(np.exp,3,3,0,2)
>>> c1; c2
poly1d([ 0.01443847, 0.128842 , 0.55284547, 0.99999962])
poly1d([-0.0049658 , 0.07610473, -0.44716929, 1. ])
x = np.linspace(0,4)
h = plt.plot(x, wp.polyval(c1,x)/wp.polyval(c2,x),'g')
h = plt.plot(x, np.exp(x), 'r')
See also
--------
padefit
Reference
---------
William H. Press, Saul Teukolsky,
William T. Wetterling and Brian P. Flannery (1997)
"Numerical recipes in Fortran 77", Vol. 1, pp 197-20
"""
def _points(npt, end_points):
if end_points:
# Use the location of the local extreme values of
# the Chebychev polynomial of the first kind of degree NPT-1.
return chebextr(npt - 1)
# Use the roots of the Chebychev polynomial of the first kind of
# degree NPT. Note this is useful if there are singularities close
# to the endpoints.
return chebroot(npt, kind=1)
def _check_size(fs, npt):
if len(fs) < npt:
warnings.warn('Check the result! Number of function values ' +
'should be at least: {0:d}'.format(npt))
NFAC = 8
BIG = 1e30
MAXIT = 5
smallest_devmax = BIG
ncof = m + k + 1
# Number of points where function is evaluated, i.e. fineness of mesh
npt = NFAC * ncof
if x is None:
x = map_to_interval(_points(npt, end_points), a, b)
if hasattr(fun, '__call__'):
fs = fun(x)
else:
fs = fun
_check_size(fs, npt)
if trace:
plt.plot(x, fs, '+')
wt = np.ones((npt))
ee = np.ones((npt))
mad = 0
u = np.zeros((npt, ncof))
for ix in range(MAXIT):
# Set up design matrix for least squares fit.
pow1 = wt
bb = pow1 * (fs + abs(mad) * np.sign(ee))
for jx in range(m + 1):
u[:, jx] = pow1
pow1 = pow1 * x
pow1 = -bb
for jx in range(m + 1, ncof):
pow1 = pow1 * x
u[:, jx] = pow1
[u1, w, v] = np.linalg.svd(u, full_matrices=False)
cof = np.dot(np.where(w == 0, 0.0, np.dot(bb, u1) / w), v)
# Tabulate the deviations and revise the weights
ee = polyval(cof[m::-1], x) / \
polyval(cof[ncof:m:-1].tolist() + [1, ], x) - fs
wt = np.abs(ee)
devmax = max(wt)
mad = wt.mean() # mean absolute deviation
# Save only the best coefficients found
if (devmax <= smallest_devmax):
smallest_devmax = devmax
c1 = cof[m::-1]
c2 = cof[ncof:m:-1].tolist() + [1, ]
if trace:
print('Iteration=%d, max error=%g' % (ix, devmax))
plt.plot(x, fs, x, ee + fs)
return poly1d(c1), poly1d(c2)
def main():
exp = np.exp
[c1, c2] = padefitlsq(exp, 3, 3, 0, 2)
x = np.linspace(0, 4)
plt.plot(x, polyval(c1, x) / polyval(c2, x), 'g')
plt.plot(x, exp(x), 'r')
import scipy.special as sp
p = [[1, 1, 1], [2, 2, 2]]
pi = polyint(p, 1)
_pr = polyreloc(p, 2)
_pd = polyder(p)
_st = poly2str(p)
# polynomial coeff exponential function:
c = poly1d(1. / sp.gamma(np.r_[6 + 1:0:-1]))
[p, q] = padefit(c)
x = np.linspace(0, 4)
plt.plot(x, c(x), x, p(x) / q(x), 'g-', x, exp(x), 'r.')
plt.close()
x = arange(4)
dx = dct(x)
_idx = idct(dx)
a = 0
b = 2
ck = chebfit(exp, 6, a, b)
_t = chebval(0, ck, a, b)
x = np.linspace(0, 2, 6)
plt.plot(x, exp(x), 'r', x, chebval(x, ck, a, b), 'g.')
# x1 = chebroot(9).'*(b-a)/2+(b+a)/2 ;
# ck1 =chebfit([x1 exp(x1)],9,a,b);
# plot(x,exp(x),'r'), hold on
# plot(x,chebval(x,ck1,a,b),'g'), hold off
_t = poly2hstr([1, 1, 2])
py = [1, 0]
px = polyshift(py, 0, 5)
_t1 = polyval(px, [0, 2.5, 5]) # % This is the same as the line below
_t2 = polyval(py, [-1, 0, 1])
px = [1, 0]
py = polyishift(px, 0, 5)
t1 = polyval(px, [0, 2.5, 5]) # % This is the same as the line below
t2 = polyval(py, [-1, 0, 1])
print(t1, t2)
def test_polydeg():
x = np.linspace(0, 10, 300)
y = np.sin(x ** 3 / 100) ** 2 + 0.05 * np.random.randn(x.size)
n = polydeg(x, y)
# n = 2
p = orthofit(x, y, n)
xi = np.linspace(x.min(), x.max())
ys0 = orthoval(p, x)
ys = orthoval(p, xi)
ys2 = orthoval(p, xi)
plt.plot(x, y, '.', x, ys0, 'k', xi, ys, 'r', xi, ys2, 'r.')
p0 = ortho2poly(p)
p1 = polyfit(x, ys0, n)
plt.plot(xi, polyval(p0, xi), 'g-.', xi, polyval(p1, xi), 'go')
plt.show('hold')
def test_docstrings():
import doctest
print('Testing docstrings in %s' % __file__)
options = doctest.NORMALIZE_WHITESPACE | doctest.ELLIPSIS
doctest.testmod(optionflags=options, verbose=False)
def chebvandernd(deg, *xi):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x1, x2, ..., xn)`. If `l, m, n` are the given degrees in
`x1, x2, x3`, then The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x1)*T_j(x2)*T_k(x3),
where `0 <= i <= l`, `0 <= j <= m`, and `0 <= k <= n`. The leading
indices of `V` index the points `(x, y, z)` and the last index encodes
the degrees of the Chebyshev polynomials.
If ``V = chebvandernd([xdeg, ydeg, zdeg], x, y, z)``, then the columns
of `V` correspond to the elements of a 3-D coefficient array `c` of
shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
.. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
and ``np.dot(V, c.flat)`` and ``chebvalnd(c, x, y, z)`` will be the
same up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of N-D Chebyshev
series of the same degrees and sample points.
Parameters
----------
deg : list of ints
List of maximum degrees of the form [x1_deg, x2_deg, ...,xn_deg].
x1, x2, ..., xn : array_like
Arrays of point coordinates, all of the same shape. The dtypes will
be converted to either float64 or complex128 depending on whether
any of the elements are complex. Scalars are converted to 1-D
arrays.
Returns
-------
vander : ndarray
The shape of the returned matrix is ``x1.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg([1]+1)*...*(deg[n-1]+1)`. The dtype
will be the same as the converted `x1`, `x2`, ... `xn`.
See Also
--------
chebvander, chebvalnd, chebfitnd
"""
ideg = [int(d) for d in deg]
is_valid = np.array([di == d and di >= 0 for di, d in zip(ideg, deg)])
if np.any(is_valid != 1):
raise ValueError("degrees must be non-negative integers")
ndim = len(xi)
if len(ideg) != ndim:
msg = 'length of deg must be the same as number of dimensions'
raise ValueError(msg)
xi = np.array(xi, copy=0) + 0.0
chebvander = np.polynomial.chebyshev.chebvander
shape0 = xi[0].shape
s0 = (1,) * ndim
vxi = [chebvander(x, d).reshape(shape0 + s0[:i] + (-1,) + s0[i + 1::])
for i, (d, x) in enumerate(zip(ideg, xi))]
v = reduce(np.multiply, vxi)
return v.reshape(v.shape[:-ndim] + (-1,))
def chebfitnd(xi, f, deg, rcond=None, full=False, w=None):
"""
Least squares fit of Chebyshev series to N-dimensional data.
Return the coefficients of a Chebyshev series of degree `deg` that is the
least squares fit to the data values `f` given at points
`x1`, `x2`,..., `xn`
The fitted polynomial(s) are in the form
.. math:: p(x,y) = c_00 + c_11 * T_1(x)*T_1(y) + ..c_ij * T_i(x)*T_j(y).
+ c_nm * T_n(x)*T_m(y),
where `n`, `m` is `deg`.
Parameters
----------
xi: tuple
x1-, x2-,....xn-coordinates of the sample points.
f : array_like
function values at the sample points ``(x1[i], x2[i], ..., xn[i])``.
deg : list
Degrees of the fitting series in the x1, x2, ..., xn directions,
respectively.
rcond : float, optional
Relative condition number of the fit. Singular values smaller than
this relative to the largest singular value will be ignored. The
default value is size(x1)*eps, where eps is the relative precision of
the float type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is False (the
default) just the coefficients are returned, when True diagnostic
information from the singular value decomposition is also returned.
w : array_like, optional
Weights. If not None, the contribution of each point
``(x1[i], x2[i], ..., xn[i])`` to the fit is weighted by `w[i]`.
Ideally the weights are chosen so that the errors of the products
``w[i]*f[i]`` all have the same variance. The default value is None.
Returns
-------
coef : ndarray, shape (M1, M2,..., Mn)
Chebyshev coefficients ordered from low to high.
[residuals, rank, singular_values, rcond] : list
These values are only returned if `full` = True
resid -- sum of squared residuals of the least squares fit
rank -- the numerical rank of the scaled Vandermonde matrix
sv -- singular values of the scaled Vandermonde matrix
rcond -- value of `rcond`.
For more details, see `linalg.lstsq`.
Warns
-----
RankWarning
The rank of the coefficient matrix in the least-squares fit is
deficient. The warning is only raised if `full` = False. The
warnings can be turned off by
>>> import warnings
>>> warnings.simplefilter('ignore', RankWarning)
See Also
--------
chebvalnd, chebgridnd
Notes
-----
The solution is the coefficients of the Chebyshev series `p` that
minimizes the sum of the weighted squared errors
.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
where :math:`w_j` are the weights. This problem is solved by setting up
as the (typically) overdetermined matrix equation
.. math:: V(x, y) * c = w * y,
where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
coefficients to be solved for, `w` are the weights, and `y` are the
observed values. This equation is then solved using the singular value
decomposition of `V`.
If some of the singular values of `V` are so small that they are
neglected, then a `RankWarning` will be issued. This means that the
coefficient values may be poorly determined. Using a lower order fit
will usually get rid of the warning. The `rcond` parameter can also be
set to a value smaller than its default, but the resulting fit may be
spurious and have large contributions from roundoff error.
Fits using Chebyshev series are usually better conditioned than fits
using power series, but much can depend on the distribution of the
sample points and the smoothness of the data. If the quality of the fit
is inadequate splines may be a good alternative.
References
----------
.. [1] Wikipedia, "Curve fitting",
http://en.wikipedia.org/wiki/Curve_fitting
Examples
--------
"""
def _check_shapes(z, ndims, sizes):
ndim = len(ndims)
if np.any(ndims != ndim) or z.ndim != ndim:
msg = "expected {0:d}D array for x1, x2,...,xn and f".format(ndim)
raise TypeError(msg)
if np.any(sizes == 0):
raise TypeError("expected non-empty vector for xi")
def _check_size(w, n):
if n != len(w):
raise TypeError("expected x and w to have same length")
# xi = np.array(xi, copy=0) + 0.0
z = np.array(f)
degrees = np.asarray(deg, dtype=int)
orders = degrees + 1
order = np.product(orders)
ndims = np.array([x.ndim for x in xi])
sizes = np.array([x.size for x in xi])
_check_shapes(z, ndims, sizes)
lhs = chebvandernd(degrees, *xi).reshape((-1, order))
rhs = z.ravel()
if w is not None:
w = np.asarray(w).ravel() + 0.0
_check_size(w, len(lhs))
lhs = lhs * w
rhs = rhs * w
if rcond is None:
rcond = xi[0].size * np.finfo(x.dtype).eps
if issubclass(lhs.dtype.type, np.complexfloating):
scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(axis=0))
else:
scl = np.sqrt(np.square(lhs).sum(axis=0))
scl[scl == 0] = 1
# Solve the least squares problem.
c, resids, rank, s = np.linalg.lstsq(lhs/scl, rhs, rcond)
c = (c/scl).reshape(orders)
if full:
return c, [resids, rank, s, rcond]
elif rank != order:
msg = "The fit may be poorly conditioned"
warnings.warn(msg, pu.RankWarning)
return c
def chebvalnd(c, *xi):
"""
Evaluate a N-D Chebyshev series at points (x1, x2, ..., xn).
This function returns the values:
.. math:: p(x1,x2,...,xn) =
\\sum_{i,j,...,k} c_{i,j,...,k} * T_i(x1) * T_j(x2)*...* T_k(xn)
The parameters `x1`, `x2`, ...., `xn` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x1`, `x2`, ..., `xn` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.
If `c` has fewer than N dimensions, ones are implicitly appended to its
shape to make it N-D. The shape of the result will be c.shape[3:] +
x1.shape.
Parameters
----------
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j,...,k is contained in ``c[i,j,...,k]``. If `c` has
dimension greater than N the remaining indices enumerate multiple sets
of coefficients.
x1, x2,..., xn : array_like, compatible object
The N dimensional series is evaluated at the points
`(x1, x2,...,xn)`, where `x1`, `x2`,..., `xn` must have the same shape.
If any of `x1`, `x2`, ..., `xn` is a list or tuple, it is first
converted to an ndarray, otherwise it is left unchanged and if it isn't
an ndarray it is treated as a scalar.
Returns
-------
values : ndarray, compatible object
The values of the multidimensional polynomial on points formed with
triples of corresponding values from `x`, `y`, and `z`.
See Also
--------
chebval, chebgridnd, chebfitnd
"""
try:
xi = np.array(xi, copy=0)
except:
raise ValueError('x, y, z are incompatible')
chebval = np.polynomial.chebyshev.chebval
c = chebval(xi[0], c)
for x in xi[1:]:
c = chebval(x, c, tensor=False)
return c
def chebgridnd(c, *xi):
"""
Evaluate a N-D Chebyshev series on the Cartesian product of x1, x2,..., xn.
This function returns the values:
.. math:: p(a,b,...) = \\sum_{i,j,...} c_{i,j,...} * T_i(a) * T_j(b) * ...
where the points `(a, b, ...)` consist of all points formed by taking
`a` from `x1`, `b` from `x2`, and so on. The resulting points form
a grid with `x1` in the first dimension, `x2` in the second, and so on.
The parameters `x1`, `x2`, ... and `xn` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars. In
either case, either `x1`, `x2`,... and `xn` or their elements must support
multiplication and addition both with themselves and with the elements
of `c`.
If `c` has fewer than N dimensions, ones are implicitly appended to
its shape to make it N-D. The shape of the result will be c.shape[3:] +
x1.shape + x2.shape + ... + xn.shape
Parameters
----------
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree i,j are contained in ``c[i,j]``. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
x1, x2,..., xn : ndarray, compatible object
1-D arrays representing the coordinates of a grid.
The N dimensional series is evaluated at the points in the
Cartesian product of `x1`, `x2`, ... and `xn`. If `xi`, is a
list or tuple, it is first converted to an ndarray, otherwise it is
left unchanged and, if it isn't an ndarray, it is treated as a
scalar.
Returns
-------
values : ndarray, compatible object
The values of the N dimensional polynomial at points in the Cartesian
product of `x1`, `x2`, ... and `xn`.
See Also
--------
chebval, chebvalnd, chebfitnd
"""
chebval = np.polynomial.chebyshev.chebval
for x in xi:
c = chebval(x, c)
return c
def test_chebfit1d():
def f(x):
return np.exp(-x**2)
# x = chebroot(n=64, kind=1)
# z = f(x)
c = chebfit(f, n=64)[::-1]
xi = np.linspace(-1, 1, 151)
zi = np.polynomial.chebyshev.chebval(xi, c)
# plt.plot(xi, zi,'.', xi, f(xi))
plt.semilogy(xi, np.abs(zi-f(xi)))
plt.show('hold')
def test_chebfit2d():
n = 3
xorder, yorder = n-1, n-1
x = chebroot(n=n, kind=1)
xgrid, ygrid = np.meshgrid(x, x)
def f(x, y):
return np.exp(-x**2-6*y**2)
zgrid = f(xgrid, ygrid)
# v2d = np.polynomial.chebyshev.chebvander2d(xgrid, ygrid,
# [xorder,yorder]).reshape((-1, (xorder+1)*(yorder+1)))
# coeff, residuals, rank, s = np.linalg.lstsq(v2d, zgrid.ravel())
# doeff = coeff.reshape(xorder+1,yorder+1)
_dcoeff2 = chebfitnd((xgrid, ygrid), zgrid, [xorder, yorder])
dcoeff = chebfit_dct(f, n=(xorder+1, yorder+1))
xi = np.linspace(-1, 1, 151)
Xi, Yi = np.meshgrid(xi, xi)
Zi = f(Xi, Yi)
zzi = chebvalnd(dcoeff, Xi, Yi)
_devi = Zi - zzi
# plot residuals
# zz = np.polynomial.chebyshev.chebval2d(xgrid, ygrid, dcoeff)
zz = chebvalnd(dcoeff, xgrid, ygrid)
dev = zgrid - zz
# plt.spy(np.abs(dcoeff)>1e-13)
plt.contourf(xgrid, ygrid, np.abs(dev))
# plt.contourf(Xi, Yi, np.abs(devi))
plt.colorbar()
# plt.semilogy(np.abs(devi.ravel()))
plt.show('hold')
if __name__ == '__main__':
if False: # True: #
main()
else:
# test_chebfit2d()
test_docstrings()
# test_polydeg()
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